#7633353

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂³ ▷ C₁

6654 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)(17,18,20,22)(19,23,24,21);(2,5)(6,8);(10,13)(14,16);(17,19,20,24)(18,21,22,23);(9,11,12,15)(10,14,13,16);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239862, #72239864, #72239872, …
maximal normal subgroup of	#72239862, #72239864, #72239872, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540549);