#7631365

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
prime radical	C₂×#2408	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7108 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=g,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72241234, #72241235, #72241246
maximal normal subgroup of	#72241234, #72241235, #72241246

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538561);