#10325231

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

5296 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:

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

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,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265951, #72265955, #72265962
maximal normal subgroup of	#72265951, #72265955, #72265962
maximal quotient of	#72266512, #72266523

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10232427);