#850537

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
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×#127	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 757733);