#92862

order	512
nilpotency class	4
isoclinism	isoclinic class of #7092
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#331
commutator	C₄×C₈	C₂×C₈
Frattini, cosocle	C₄²×C₈	C₂²
socle	C₂²	#1270
prime radical	#3169	C₄
LCS	G ▷ C₄×C₈ ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂×C₄² ▷ C₂² ▷ C₁

220 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 52 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) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2)(3,10,11,16,12,15,13,6)(4,7)(5,9)(8,14)(17,18)(19,22)(20,29)(21,25)(23,26)(24,27)(28,31)(30,32);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14)(18,22)(19,26)(20,27)(23,24)(25,31)(28,32)(29,30)

Polycyclic presentation:

2^9|a^2=d,c^2=fi,d^2=g,e^2=h,f^2=i,b^a=bc,c^a=ce,e^a=eh,f^a=fh,c^b=cf,d^b=defh,e^b=eh,f^b=fi,g^b=ghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227208
maximal normal subgroup of	#72227208
maximal quotient of	#72227205

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58);