#414986

order	512
nilpotency class	3
isoclinism	isoclinic class of #393881
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

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

1698 conjugacy classes of subgroups.

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=efhi,b^2=i,d^2=h,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322182);