#6552651

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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
prime radical	C₂×#3318	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2005 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308641, #72308845
maximal normal subgroup of	#72308641, #72308845

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6459847);