#774062

order	512
nilpotency class	2
isoclinism	isoclinic class of #772778
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

4484 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4)(6,8)(13,14,17,16)(15,19,20,18);(2,4)(9,10,11,12);(9,10)(11,12)(13,15,17,20)(14,18,16,19);(2,4)(5,6)(7,8)(13,16,17,14)(15,19,20,18);(1,2)(3,4)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281141, #72281436, #72281552
maximal normal subgroup of	#72281141, #72281436, #72281552

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 681258);