#4362970

order	512
nilpotency class	2
isoclinism	isoclinic class of #4311730
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3540 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279892
maximal normal subgroup of	#72279892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4270166);