#6630263

order	512
nilpotency class	3
isoclinism	isoclinic class of #32929
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	#32930	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286189, #72286209, #72286217, …
maximal normal subgroup of	#72286189, #72286209, #72286217, …
maximal quotient of	C₄×#32930

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6537459);