#412169

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60643	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1542 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)(7,19)(8,13)(9,24)(10,17)(11,18)(12,16)(14,22)(15,27)(20,28)(21,29)(23,36)(25,32)(26,31)(30,34)(33,35);(1,2,4,3)(5,7,10,8,11,21,12,22)(6,13,16,19,18,14,17,29)(9,20,25,23,26,34,27,35)(15,28,31,33,32,30,24,36);(1,3,4,2)(5,8,10,21,11,22,12,7)(6,14,16,29,18,13,17,19)(9,23,25,34,26,35,27,20)(15,30,31,36,32,28,24,33);(5,9,12,27,11,26,10,25)(6,15,17,24,18,32,16,31)(7,20,22,35,21,34,8,23)(13,28,29,36,14,30,19,33)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224916, #72224952, #72226819
maximal normal subgroup of	#72224916, #72224952, #72226819
maximal quotient of	#72224910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319365);