#7630239

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5504 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gi,e^2=i,f^2=gh,b^a=bg,c^a=ch,c^b=ci,e^b=ei,d^c=di,e^d=ei

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537435);