#651991

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3056 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)(5,6,9,10)(7,11,12,8)(13,14)(15,19)(16,20)(17,18)(21,22)(23,25)(24,26)(27,28);(6,10)(8,11)(14,18)(15,20)(22,23)(24,28);(13,15,17,20)(14,19,18,16)(21,23,25,22)(24,27,28,26);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16,17,19)(14,20,18,15);(5,8)(6,7)(9,11)(10,12)(13,14)(15,19)(16,20)(17,18)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|a^2=fh,c^2=h,d^2=gh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 559187);