#7630096

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5974 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gi,f^2=h,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.

Relationships

maximal subgroup of	#72304318, #72304334, #72304492
maximal normal subgroup of	#72304318, #72304334, #72304492

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537292);