#93190

order	512
nilpotency class	4
isoclinism	isoclinic class of #93181
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#126	C₂²
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#126 ◁ G
LECS	G ▷ C₂²×#126 ▷ C₂⁴ ▷ C₂² ▷ C₁

324 conjugacy classes of subgroups; only the 40 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 40 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 64 points:

(1,2,5,11,8,15,31,47)(3,16,18,49,20,53,58,37)(4,22,24,45,26,42,35,48)(6,32,25,10,28,12,29,43)(7,13,30,46,34,14,33,41)(9,19,36,17,40,54,44,23)(21,52,55,63,59,51,27,62)(38,57,64,61,39,60,50,56);(1,3,6,19,34,59,24,57)(2,9,12,37,14,38,48,63)(4,23,5,27,29,56,33,18)(7,21,35,60,8,20,28,54)(10,16,41,50,42,51,11,44)(13,39,45,62,15,40,32,49)(17,31,55,25,61,30,58,26)(22,52,47,36,43,53,46,64)

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,e^2=hi,b^a=bc,c^a=cf,e^a=egh,f^a=fh,g^a=gi,c^b=cg,d^b=dfh,f^b=fi,g^b=ghi,d^c=dh,e^c=ehi,e^d=ei

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72235802, #72235818, #72235833, …
maximal normal subgroup of	#72235802, #72235818, #72235833, …
maximal quotient of	#72237329, #72237330

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 386);