#7151446

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#30985	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1866 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,c^2=i,d^2=fi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7058642);