#7021758

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2298 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:

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

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=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dh,e^c=ei,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, 6928954);