#213779

order	512
nilpotency class	3
isoclinism	isoclinic class of #213778
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

897 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 897 conjugacy classes of subgroups — too many to draw. Only its 210 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=ce,f^b=fh,g^b=gh,d^c=di,f^c=fh,g^c=gh,g^d=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 120975);