#2122071

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

3019 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)(5,6,9,11)(7,8,12,10)(14,17)(15,19)(22,25)(23,27);(5,7,9,12)(6,10,11,8)(14,17)(18,20)(21,22)(23,26)(24,25)(27,28);(6,11)(7,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28);(1,2)(3,4)(5,8,9,10)(6,12,11,7)(13,14)(15,18)(16,17)(19,20)(21,22)(23,26)(24,25)(27,28);(5,6)(7,10)(8,12)(9,11)(13,15)(14,18)(16,19)(17,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=gi,b^2=fgh,c^2=gh,d^2=fgh,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.

Relationships

maximal subgroup of	#72281311, #72281476, #72281741
maximal normal subgroup of	#72281311, #72281476, #72281741

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2029267);