#719632

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

2408 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)(7,12)(8,11)(9,10)(13,14,18,17)(15,20,16,19)(22,25)(26,28);(6,10)(8,12)(13,15,18,16)(14,19,17,20);(6,10)(7,11)(15,16)(19,20)(21,22)(23,26)(24,25)(27,28);(1,2,3,4)(5,7,9,11)(6,8,10,12)(13,16,18,15)(14,19,17,20)(21,22)(23,26)(24,25)(27,28);(5,8)(6,11)(7,10)(9,12)(13,17,18,14)(15,19,16,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,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, 626828);