#365456

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1443 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,18)(8,23)(9,26)(10,29)(11,16)(12,17)(13,20)(14,19)(15,24)(21,32)(22,31)(25,33)(27,36)(28,35)(30,34);(1,3,4,2)(5,7)(6,13)(8,24)(9,19)(10,21)(11,20)(12,22)(14,15)(16,31)(17,32)(18,29)(23,35)(25,27)(26,33)(28,30)(34,36);(5,8,10,19,12,25,11,30)(6,14,16,23,17,34,29,33)(7,9,21,27,22,28,20,24)(13,26,31,15,32,35,18,36);(5,9,12,28)(6,15,17,36)(7,19,22,30)(8,21,25,20)(10,27,11,24)(13,14,32,34)(16,35,29,26)(18,33,31,23)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,g^b=gi,g^c=gi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 272652);