#368864

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1722 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)(7,17)(8,19)(9,25)(10,28)(11,16)(12,20)(13,24)(14,26)(15,31)(18,30)(21,33)(22,29)(23,32)(27,34)(35,36);(1,3,4,2)(5,7)(6,12)(8,14)(9,19)(10,29)(11,21)(13,27)(15,33)(16,22)(17,28)(18,24)(20,31)(23,34)(25,35)(26,32)(30,36);(5,8)(6,13,16,19)(7,18,21,14)(9,22)(10,23)(11,24)(12,27)(15,32,28,36)(17,30)(20,34,29,25)(26,33)(31,35);(5,9,11,27)(6,14,16,18)(7,19,21,13)(8,22,24,12)(10,30,31,26)(15,34,28,25)(17,35,33,23)(20,32,29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314443, #72314450, #72314471
maximal normal subgroup of	#72314443, #72314450, #72314471
maximal quotient of	#72314514

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276060);