#778472

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2102 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)(7,11)(8,9)(10,12)(14,18)(15,20)(21,22,26,28)(23,24,25,27);(5,7,10,8)(6,9,12,11)(13,14)(15,16)(17,18)(19,20)(21,23,26,25)(22,24,28,27);(5,7,10,8)(6,11,12,9)(13,15)(14,19)(16,18)(17,20)(21,24,26,27)(22,23,28,25);(1,2)(3,4)(5,8,10,7)(6,11,12,9)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)(25,27)(26,28);(5,9,10,11)(6,8,12,7)(13,16)(14,20)(15,18)(17,19)(21,25,26,23)(22,27,28,24)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fg,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, 685668);