#368140

order	512
nilpotency class	3
isoclinism	isoclinic class of #23892
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
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

1559 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=g,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,f^c=fi,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, 275336);