#611301

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2450 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)(5,6,8,10)(7,11,12,9)(21,22)(23,28)(24,27)(25,26);(13,14)(15,18)(16,17)(19,20)(22,26)(27,28);(5,7,8,12)(6,9,10,11)(14,17)(18,20)(21,22)(23,28)(24,27)(25,26);(2,4)(13,15,16,19)(14,18,17,20)(21,23,25,24)(22,27,26,28);(21,24,25,23)(22,27,26,28)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281440, #72281453, #72282088
maximal normal subgroup of	#72281440, #72281453, #72282088

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 518497);