#774691

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3468 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)(5,6,9,10)(7,8,11,12)(13,14)(15,20)(16,19)(17,18)(21,22)(23,26)(24,27)(25,28);(6,10)(8,12)(14,18)(19,20)(21,23,24,28)(22,25,27,26);(13,15,17,16)(14,19,18,20)(21,22)(23,26)(24,27)(25,28);(1,2)(3,4)(5,7)(6,8)(9,11)(10,12)(13,16,17,15)(14,19,18,20)(21,22)(23,25)(24,27)(26,28);(5,8)(6,11)(7,10)(9,12)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,28)(25,27)

Polycyclic presentation:

2^9|a^2=fgh,b^2=h,c^2=g,d^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, 681887);