#7149092

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#44882	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1999 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056288);