#846595

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

3558 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)(5,6,8,9)(7,11,12,10)(13,14,16,18)(15,17,19,20)(21,22,25,26)(23,24,27,28);(6,9)(10,11)(13,14,16,18)(15,17,19,20)(22,26)(23,27);(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20)(21,23)(22,24)(25,27)(26,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,17)(16,18)(19,20);(5,7,8,12)(6,10,9,11)(13,15,16,19)(14,17,18,20)(21,24)(22,23)(25,28)(26,27)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262813, #72262820, #72262824, …
maximal normal subgroup of	#72262813, #72262820, #72262824, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 753791);