#93187

order	512
nilpotency class	4
isoclinism	isoclinic class of #93181
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂³×C₄	C₄²
Frattini, cosocle	C₂²×#123	C₂²
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×#123 ▷ C₂⁴ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 421 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=h,d^2=i,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	#72235162, #72235170, #72235171
maximal normal subgroup of	#72235162, #72235170, #72235171

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383);