#845007

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2608 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)(5,6,8,10)(7,9,11,12)(13,14)(15,20)(16,18)(17,19)(21,22,25,27)(23,26,28,24);(5,7)(6,9)(8,11)(10,12)(13,15)(14,18)(16,17)(19,20)(21,23)(22,26)(24,27)(25,28);(7,11)(9,12)(13,16,17,15)(14,18,19,20)(22,27)(23,28);(2,4)(15,16)(18,20)(22,27)(23,28);(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

2^9|a^2=fh,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=dfg,d^c=df

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, 752203);