#7758294

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×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5134 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:

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

Polycyclic presentation:

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

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, 7665490);