#7614450

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45271	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1928 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)(7,11)(8,12)(9,10)(13,14,19,23)(15,25,17,21)(16,18,24,20)(22,26,27,28);(2,4)(6,10)(11,12)(13,15,19,17)(14,21,23,25)(16,27,24,22)(18,26,20,28);(5,7,9,8)(6,11,10,12)(13,16,20,14,19,24,18,23)(15,25,26,22,17,21,28,27);(1,2,3,4)(13,17,20,28,19,15,18,26)(14,22,24,21,23,27,16,25);(5,8,9,7)(6,11,10,12)(13,18,19,20)(14,16,23,24)(15,26,17,28)(21,27,25,22)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fgi,d^2=h,e^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7521646);