#773540

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3073 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 680736);