#2228210

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2622 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)(7,11)(8,12)(9,10)(14,18)(15,19)(21,22,24,26)(23,27,28,25);(1,2,3,4)(5,7)(6,8)(9,12)(10,11)(13,14,17,18)(15,20,19,16)(21,22)(23,27)(24,26)(25,28);(1,2)(3,4)(6,10)(7,12)(14,18)(16,20)(21,23,24,28)(22,25,26,27);(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20);(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,20)(18,19)(21,22)(23,27)(24,26)(25,28)

Polycyclic presentation:

2^9|a^2=fi,b^2=gh,c^2=fi,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, 2135406);