#378183

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#121
prime radical	#23438	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1389 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 285379);