#794471

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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

1830 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=gi,c^2=gh,d^2=fh,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^d=ef

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 701667);