#1128813

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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	C₂×#2539	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3113 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=fg,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281853, #72281908, #72282061
maximal normal subgroup of	#72281853, #72281908, #72282061

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1036009);