#837024

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3146 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)(6,9)(10,12)(13,14)(15,19)(16,18)(17,20)(21,22)(23,28)(24,26)(25,27);(13,14,16,18)(15,17,20,19)(21,23)(22,24)(25,26)(27,28);(1,2)(3,4)(21,22)(23,28)(24,26)(25,27);(5,6)(7,10)(8,9)(11,12)(13,15)(14,17)(16,20)(18,19)(21,22)(23,24)(25,27)(26,28);(5,7,8,11)(6,10,9,12)(13,14)(15,19)(16,18)(17,20)(21,24)(22,26)(23,27)(25,28)

Polycyclic presentation:

2^9|b^2=fh,e^2=fhi,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, 744220);