#609723

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=df

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304050, #72304078, #72304116
maximal normal subgroup of	#72304050, #72304078, #72304116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 516919);