#2117335

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	#15630	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2593 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)(7,11)(8,9)(10,12)(14,18)(15,19)(21,22)(23,28)(24,27)(25,26);(5,7)(6,10)(8,11)(9,12)(13,14,17,18)(15,20,19,16)(21,23)(22,24)(25,28)(26,27);(6,8)(10,11)(14,18)(16,20)(22,25)(23,27);(1,2)(3,4)(5,7)(6,10)(8,11)(9,12)(13,15)(14,16)(17,19)(18,20)(21,24,26,28)(22,27,25,23);(5,8,9,6)(7,11,12,10)(13,16)(14,15)(17,20)(18,19)(21,25,26,22)(23,28,27,24)

Polycyclic presentation:

2^9|a^2=ghi,b^2=fg,d^2=fgh,e^2=g,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2024531);