#2088330

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3481 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:

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

Polycyclic presentation:

2^9|a^2=fg,c^2=fh,d^2=fh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327161, #72327251, #72327274
maximal normal subgroup of	#72327161, #72327251, #72327274
maximal quotient of	#72328797

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1995526);