#314642

order	512
nilpotency class	3
isoclinism	isoclinic class of #251818
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2482
prime radical	#3455	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

480 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,12)(4,11)(5,10)(6,9)(7,14)(8,13)(15,16)(17,18,23,29,21,27,22,28)(19,26,32,45,31,24,38,40)(20,33,36,25,37,43,35,44)(30,46,34,39,47,41,48,42);(1,3)(2,7)(4,15)(5,14)(6,13)(8,11)(9,16)(10,12)(17,19,21,31)(18,24,27,26)(20,34)(22,38,23,32)(25,41)(28,45,29,40)(30,35)(33,39)(36,47)(37,48)(42,43)(44,46);(1,3)(2,8)(4,13)(5,14)(6,15)(7,11)(9,12)(10,16)(17,20)(18,25)(19,30)(21,37)(22,36)(23,35)(24,39)(26,42)(27,44)(28,43)(29,33)(31,47)(32,48)(34,38)(40,46)(41,45);(1,4,5,6)(2,9,10,11)(3,13,14,15)(7,8,12,16)(17,19)(18,26)(20,30)(21,31)(22,38)(23,32)(24,27)(25,42)(28,40)(29,45)(33,41)(34,36)(35,48)(37,47)(39,44)(43,46)

Polycyclic presentation:

2^9|a^2=g,b^2=ei,d^2=i,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,d^b=dhi,f^b=fh,e^c=eh,f^c=fhi,g^c=gh,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 221838);