#375302

order	512
nilpotency class	3
isoclinism	isoclinic class of #24274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

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×#3150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1712 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216194
maximal normal subgroup of	#72216194
maximal quotient of	#72215286

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282498);