#6782895

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#23853	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1642 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2)(3,7)(4,6)(5,8)(17,18)(19,26)(20,22)(21,25)(23,31)(24,30)(27,28)(29,32);(1,3)(2,5)(4,8)(6,7)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,24,29,27)(21,31,23,25)(26,28,32,30);(10,13)(14,16)(17,19,23,30,22,29,21,28)(18,24,25,26,20,27,31,32);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,27,29,24)(21,25,23,31)(26,30,32,28);(3,8)(5,7)(10,13)(14,16)(17,21,22,23)(18,25,20,31)(19,28,29,30)(24,26,27,32)

Polycyclic presentation:

2^9|b^2=i,c^2=g,d^2=hi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240756, #72257577, #72264215
maximal normal subgroup of	#72240756, #72257577, #72264215
maximal quotient of	#72240709

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6690091);