#868723

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15124	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3060 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=f,c^2=hi,d^2=gh,e^2=f,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, 775919);