#1606431

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×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

3196 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)(5,6)(7,11)(8,9)(10,12)(13,14,17,18)(15,19,20,16)(21,22,24,26)(23,25,28,27);(6,9)(7,12)(14,18)(16,19)(21,22)(23,27)(24,26)(25,28);(5,6,8,9)(7,10,12,11)(13,15,17,20)(14,19,18,16)(21,22,24,26)(23,25,28,27);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,22)(23,25)(24,26)(27,28);(5,7,8,12)(6,10,9,11)(13,16)(14,15)(17,19)(18,20)(21,23,24,28)(22,25,26,27)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280171, #72280697
maximal normal subgroup of	#72280171, #72280697

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1513627);