#7199140

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 ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1933 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)(7,10)(8,11)(9,12)(13,14,18,22,19,23,17,25)(15,24,26,28,27,16,21,20);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,22)(17,26)(18,21)(19,27)(23,28)(24,25);(1,2,3,4)(5,7,10,6)(8,11,9,12)(13,16,19,24)(14,21,23,26)(15,25,27,22)(17,28,18,20);(5,8,10,9)(6,11,7,12)(13,14,18,22,19,23,17,25)(15,16,26,20,27,24,21,28);(5,9,10,8)(6,11,7,12)(13,17,19,18)(14,22,23,25)(15,21,27,26)(16,20,24,28)

Polycyclic presentation:

2^9|a^2=f,c^2=gi,d^2=fhi,e^2=h,f^2=i,b^a=bf,c^a=cg,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, 7106336);