#655558

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3653 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)(5,6,9,10)(7,11,12,8)(13,14,17,19)(15,16,20,18)(21,22)(23,27)(24,28)(25,26);(6,10)(8,11)(13,15,17,20)(14,18,19,16)(22,26)(24,27);(13,16)(14,15)(17,18)(19,20)(21,22)(23,27)(24,28)(25,26);(1,2)(3,4)(5,7)(6,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,20)(21,23,25,28)(22,27,26,24);(5,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,20)(18,19)(21,24,25,27)(22,28,26,23)

Polycyclic presentation:

2^9|a^2=fh,b^2=g,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 562754);