#608051

order	512
nilpotency class	2
isoclinism	isoclinic class of #14715
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3134 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304056, #72304093, #72304116
maximal normal subgroup of	#72304056, #72304093, #72304116

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 515247);