#238075

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
socle	C₂²	C₂×#393
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

928 conjugacy classes of subgroups; only the 178 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 178 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=i,d^2=h,b^a=be,c^a=cf,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145271);