#6439013

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438997
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

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 ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3406 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=f,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=dhi,e^b=ei,e^c=ei,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249971, #72257002, #72257012
maximal normal subgroup of	#72249971, #72257002, #72257012
maximal quotient of	#72265626

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6346209);