#7054054

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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	#44884	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2160 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)(5,6)(7,9)(8,10)(11,12)(14,18)(15,19)(20,24)(21,23)(22,27)(25,28);(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,18)(15,21,23,19)(16,24,26,20)(22,27,25,28);(5,7,8,12)(6,9,10,11)(13,15,17,23)(14,19,18,21)(16,25,26,22)(20,28,24,27);(2,4)(7,12)(9,11)(13,16,17,26)(14,20,18,24)(15,22,23,25)(19,27,21,28);(6,10)(9,11)(13,14,17,18)(15,19,23,21)(16,24,26,20)(22,28,25,27)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6961250);