#610030

order	512
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂⁴
commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15278	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3486 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)(6,10)(11,12)(13,14)(15,18)(16,17)(19,20)(21,22,24,26)(23,25,27,28);(5,6)(7,12)(8,11)(9,10)(14,17)(15,19)(21,23)(22,25)(24,27)(26,28);(14,17)(18,20)(23,27)(25,28);(2,4)(5,7,9,8)(6,11,10,12)(13,15,16,19)(14,18,17,20);(5,8,9,7)(6,11,10,12)

Polycyclic presentation:

2^9|a^2=hi,d^2=f,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	#72281440, #72281476, #72281496, …
maximal normal subgroup of	#72281440, #72281476, #72281496, …
maximal quotient of	#72298465

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 517226);