#7715997

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	C₂²×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7623193);