#1738295

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
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

3091 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)(6,9)(7,11)(13,14,16,17)(15,19,20,18)(22,25)(23,27);(6,9)(10,12)(14,17)(15,20)(21,22,24,25)(23,28,27,26);(5,6)(7,10)(8,9)(11,12)(13,15,16,20)(14,18,17,19)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,20)(17,19)(21,22)(23,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,27)(22,26,25,28)

Polycyclic presentation:

2^9|a^2=fgh,b^2=gh,c^2=f,e^2=fh,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	#72278311
maximal normal subgroup of	#72278311
maximal quotient of	#72278341, #72299666, #72299683

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1645491);