#368090

order	512
nilpotency class	3
isoclinism	isoclinic class of #23882
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1666 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)(5,6,12,18)(7,13,9,15)(8,21,24,33)(10,17,11,16)(14,26,32,25)(19,30,23,36)(20,34,27,29)(22,28,35,31);(1,3,4,2)(5,7,12,9)(6,13,18,15)(8,22,24,35)(10,20,11,27)(14,30,32,36)(16,29,17,34)(19,25,23,26)(21,31,33,28);(5,8,11,26,12,24,10,25)(6,14,17,33,18,32,16,21)(7,19,20,22,9,23,27,35)(13,28,29,30,15,31,34,36);(5,9,12,7)(6,15,18,13)(8,23,24,19)(10,20,11,27)(14,31,32,28)(16,29,17,34)(21,30,33,36)(22,26,35,25)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72316343, #72316345, #72316348
maximal normal subgroup of	#72316343, #72316345, #72316348

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275286);