#1422915

order	512
nilpotency class	2
isoclinism	isoclinic class of #1418353
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2340 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)(5,6,10,9)(7,8,12,11)(13,14)(15,16)(17,18)(19,20)(22,25)(23,27);(5,7,10,12)(6,11,9,8)(13,15,18,20)(14,19,17,16);(13,16)(14,20)(15,17)(18,19)(21,22)(23,26)(24,25)(27,28);(1,2)(3,4)(5,8)(6,7)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,23,24,27)(22,26,25,28);(5,9,10,6)(7,11,12,8)(13,17,18,14)(15,19,20,16)(21,22)(23,26)(24,25)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1330111);