#377365

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₂²⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=f,d^2=eh,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216212
maximal normal subgroup of	#72216212
maximal quotient of	#72215970

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284561);