#356749

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

(1,2)(3,4)(10,14,17,16)(15,21,22,23);(5,6,7,8)(9,10)(11,15)(12,16)(13,17)(14,20)(18,22)(19,23)(21,24);(2,4)(9,11,12,19,13,18,20,24)(10,15,16,23,17,22,14,21);(10,16,17,14)(11,18)(15,21,22,23)(19,24)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=eg,d^2=g,e^2=i,b^a=be,c^a=cf,d^b=de,e^b=ei,g^b=gi,d^c=di

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	D₄×#1773, #72310129, #72310142
maximal normal subgroup of	D₄×#1773, #72310129, #72310142

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263945);