#748897

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
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

3019 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)(13,14,17,16)(15,19,20,18)(22,26)(23,27);(14,16)(18,19)(21,22)(23,24)(25,26)(27,28);(1,2,3,4)(15,20)(18,19)(22,26)(23,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,20)(14,18,16,19)(21,23,25,27)(22,24,26,28);(5,7,8,11)(6,10,9,12)(13,16,17,14)(15,19,20,18)(21,24)(22,27)(23,26)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72329619, #72329666, #72329706
maximal normal subgroup of	#72329619, #72329666, #72329706
maximal quotient of	#72329893

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 656093);