#7629993

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

6219 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,22)(20,24)(21,23);(2,5)(6,8)(17,19,21,20)(18,22,23,24);(10,13)(14,16)(18,23)(19,20);(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,23,22);(19,20)(22,24);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|b^2=i,d^2=h,f^2=g,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	#72280931, #72281003, #72303638, …
maximal normal subgroup of	#72280931, #72281003, #72303638, …
maximal quotient of	#72281942, #72303671

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7537189);