#157141

order	512
nilpotency class	2
isoclinism	isoclinic class of #156684
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

3020 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=g,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267474, #72267504, #72267505, …
maximal normal subgroup of	#72267474, #72267504, #72267505, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 64337);