#8105999

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319623, #72319780
maximal normal subgroup of	#72319623, #72319780

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8013195);