#7631328

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	#36214	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=g,f^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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538524);