#386378

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=efg,e^2=i,b^a=be,c^a=cf,d^a=def,e^a=ei,c^b=cg,d^b=de,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,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, 293574);