#2137134

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2006 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)(7,10)(8,9)(11,12)(13,14,18,19)(15,20,16,17)(21,22,27,25)(23,28,26,24);(5,6,9,8)(7,10,12,11)(14,19)(15,16)(21,23,27,26)(22,28,25,24);(6,8)(10,11)(13,15,18,16)(14,20,19,17)(21,24)(22,26)(23,25)(27,28);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,16,18,15)(14,20,19,17)(21,25,27,22)(23,28,26,24);(5,8,9,6)(7,11,12,10)(13,17,18,20)(14,15,19,16)(21,26,27,23)(22,28,25,24)

Polycyclic presentation:

2^9|a^2=fhi,b^2=g,c^2=gh,d^2=h,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2044330);