#7062893

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	#25117	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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)(5,6)(7,9)(8,10)(11,12)(13,14)(15,24)(16,21)(17,27)(18,22)(19,26)(20,25)(23,28);(5,6)(7,11)(8,10)(9,12)(13,15,18,25)(14,19,22,16)(17,21,28,26)(20,27,24,23);(2,4)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,27)(17,26)(18,25)(19,23)(21,28)(22,24);(7,12)(9,11)(13,16,18,19)(14,21,22,26)(15,23,25,27)(17,24,28,20);(6,10)(9,11)(14,22)(15,25)(21,26)(23,27)

Polycyclic presentation:

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

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, 6970089);