#371021

order	512
nilpotency class	3
isoclinism	isoclinic class of #23996
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1440 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)(7,11,17,23)(8,12)(9,13,20,27)(10,14)(15,24,29,21)(16,30)(18,25,32,35)(19,26)(22,31)(28,33)(34,36);(1,3,4,2)(5,7)(6,11,14,23)(8,15)(9,16)(10,17)(12,24,26,21)(13,22,27,30)(18,28)(19,29)(20,31)(25,34,35,33)(32,36);(5,8,10,19)(6,12,14,26)(7,15,17,29)(9,18,20,32)(11,21,23,24)(13,25,27,35)(16,28,31,36)(22,33,30,34);(5,9)(6,13,14,27)(7,16)(8,18)(10,20)(11,22,23,30)(12,25,26,35)(15,28)(17,31)(19,32)(21,33,24,34)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285042, #72285164, #72285204, …
maximal normal subgroup of	#72285042, #72285164, #72285204, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 278217);