#615756

order	512
nilpotency class	2
isoclinism	isoclinic class of #612424
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3430 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)(6,10)(7,8)(14,17)(15,19)(22,24)(26,27);(5,6)(7,11)(8,12)(9,10)(13,14)(15,18)(16,17)(19,20)(21,22,25,24)(23,27,28,26);(1,2)(3,4)(5,7,9,8)(6,11,10,12)(13,14)(15,18)(16,17)(19,20)(21,23,25,28)(22,26,24,27);(5,6)(7,11)(8,12)(9,10);(5,8,9,7)(6,12,10,11)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,27,28,26)

Polycyclic presentation:

2^9|b^2=h,c^2=hi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257953, #72257957, #72257963, …
maximal normal subgroup of	#72257953, #72257957, #72257963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 522952);