#2044085

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1624 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,8)(4,7)(5,6)(10,11)(14,15)(17,18,21,20)(19,22,23,24)(25,26,30,28)(27,31,29,32);(1,3,6,4)(2,7,5,8)(9,10,13,11)(12,15,16,14)(19,23)(22,24)(25,27,30,29)(26,31,28,32);(1,4,6,3)(2,7,5,8)(9,11,13,10)(12,15,16,14)(17,19,21,23)(18,22,20,24)(25,28,30,26)(27,31,29,32);(9,12)(10,14)(11,15)(13,16)(17,20,21,18)(19,24,23,22)(25,29,30,27)(26,31,28,32);(1,5,6,2)(3,7,4,8)(9,12)(10,14)(11,15)(13,16)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=fh,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=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, 1951281);