#845681

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	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,d^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, 752877);