#1337317

order	512
nilpotency class	2
isoclinism	isoclinic class of #1258513
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	C₂×#2678	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3153 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^d=ef

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333955, #72333992, #72334021
maximal normal subgroup of	#72333955, #72333992, #72334021

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1244513);