#845009

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
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	#17086	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2906 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)(5,6,9,11)(7,12,8,10)(13,14,16,18)(15,17,19,20)(21,22,25,27)(23,26,28,24);(5,7,9,8)(6,10,11,12)(13,15)(14,17)(16,19)(18,20)(21,23)(22,26)(24,27)(25,28);(5,8,9,7)(6,10,11,12)(15,19)(17,20)(22,27)(23,28);(2,4)(7,8)(10,12)(22,27)(23,28);(21,24,25,26)(22,23,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304057, #72304096, #72304110
maximal normal subgroup of	#72304057, #72304096, #72304110

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752205);