#607950

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2674 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)(7,11)(8,10)(9,12)(13,14,16,18)(15,19,20,17);(5,7,9,8)(6,10,12,11)(21,22)(23,26)(24,25)(27,28);(5,7,9,8)(6,11,12,10)(13,15,16,20)(14,17,18,19)(22,25)(26,28);(2,4)(5,8,9,7)(6,10,12,11);(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=fg,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515146);