#385529

order	512
nilpotency class	3
isoclinism	isoclinic class of #30437
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1151 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 292725);