#8905195

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

5246 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265955, #72265961, #72265981
maximal normal subgroup of	#72265955, #72265961, #72265981
maximal quotient of	#72266539, #72266565

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8812391);