#376140

order	512
nilpotency class	3
isoclinism	isoclinic class of #23995
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1568 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285083, #72285571, #72285631
maximal normal subgroup of	#72285083, #72285571, #72285631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283336);