#7502833

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂³	C₂³×D₄
prime radical	#30417	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1952 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|b^2=i,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dfg,g^a=gi,c^b=ch,d^b=dfi,g^c=gi,g^d=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7410029);