#2032198

order	512
nilpotency class	2
isoclinism	isoclinic class of #2031997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#2670	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4308 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)(6,8)(13,14,17,18)(15,21,23,19)(16,20,24,25)(22,27,28,26);(1,2,3,4)(10,12)(14,18)(16,24)(19,21)(22,28);(9,10)(11,12)(13,15,17,23)(14,19,18,21)(16,22,24,28)(20,26,25,27);(5,6)(7,8)(13,14)(15,19)(16,20)(17,18)(21,23)(22,26)(24,25)(27,28);(1,2)(3,4)(13,16)(14,20)(15,22)(17,24)(18,25)(19,26)(21,27)(23,28)

Polycyclic presentation:

2^9|a^2=g,b^2=fg,c^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72267407
maximal normal subgroup of	#72267407
maximal quotient of	#72267461

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1939394);