#7021822

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
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	#59066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2258 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)(13,14,17,21,18,22,16,20)(15,23,24,28,26,25,27,19);(5,6)(7,11)(8,10)(9,12)(13,15,18,26)(14,19,22,28)(16,24,17,27)(20,23,21,25);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,16,18,17)(14,20,22,21)(15,24,26,27)(19,23,28,25);(7,12)(9,11)(13,14,16,20,18,22,17,21)(15,25,27,28,26,23,24,19);(6,10)(9,11)(13,16,18,17)(14,21,22,20)(15,24,26,27)(19,25,28,23)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=gi,d^2=fi,e^2=i,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^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, 6929018);