#7667009

order	512
nilpotency class	2
isoclinism	isoclinic class of #58990
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	#59041	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5832 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312186, #72341410, #72341414
maximal normal subgroup of	#72312186, #72341410, #72341414

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7574205);