#6500400

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500385
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3099 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)(13,14,18,19)(15,16,20,17);(5,6)(7,11)(8,10)(9,12)(13,15,17,14,18,20,16,19);(2,4)(5,7)(6,9)(8,12)(10,11)(14,19)(15,20);(7,12)(9,11)(13,16,18,17)(14,20,19,15);(6,10)(9,11)(13,16,18,17)(14,20,19,15)

Polycyclic presentation:

2^9|a^2=i,b^2=f,d^2=i,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=eh,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282559, #72283306
maximal normal subgroup of	#72282559, #72283306

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6407596);