#7065149

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996719
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	#24450	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1903 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6972345);