#7629217

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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	#35761	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5807 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)(11,13)(12,14)(15,16);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,18)(19,22)(20,21)(23,24);(10,14)(13,16)(18,21)(22,24);(17,19)(18,22)(20,23)(21,24);(19,23)(22,24);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72234869, #72234874, #72234896, …
maximal normal subgroup of	#72234869, #72234874, #72234896, …
maximal quotient of	C₂×#7629217

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7536413);