#356507

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1872 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312202, #72312215, #72312238
maximal normal subgroup of	#72312202, #72312215, #72312238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263703);