#794208

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2608 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)(7,12)(8,10)(9,11)(14,17)(18,20)(21,22)(23,27)(24,26)(25,28);(5,7)(6,8)(9,10)(11,12)(14,17)(18,20)(21,23,26,25)(22,27,24,28);(1,2)(3,4)(5,8,9,12)(6,10,11,7)(13,14,16,17)(15,18,19,20)(21,24,26,22)(23,27,25,28);(5,7)(6,8)(9,10)(11,12)(13,15,16,19)(14,18,17,20)(21,25,26,23)(22,27,24,28);(5,6)(7,12)(8,10)(9,11)

Polycyclic presentation:

2^9|a^2=gh,b^2=fi,c^2=fh,d^2=h,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, 701404);