#94852

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄
automorphism	?
homology	Abelianization (H₁) = C₄×C₈² · Schur multiplier (H₂) = C₄×C₈²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²×C₈	C₂²
commutator	C₂	C₄×C₈²
Frattini, cosocle	C₄³	C₂³
socle, prime radical	C₂³	C₄³
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

594 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,f^2=i,c^b=cg

Conjugacy classes & character table

320 conjugacy classes — view the full character table.

Relationships

Frattini subgroup of	#72234262
maximal subgroup of	#72210241, #72210252, C₂×#94852, …
maximal normal subgroup of	#72210241, #72210252, C₂×#94852, …
maximal quotient of	C₂×#94852, #72210602, #72234228

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2048);