#366364

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1945 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)(8,23)(9,27)(10,15)(11,16)(12,20)(13,25)(14,24)(18,34)(19,29)(21,31)(22,30)(26,33)(28,36)(32,35);(1,3,4,2)(5,7,10,20,11,22,9,21)(6,12,15,30,16,31,27,17)(8,19,24,36,26,35,25,18)(13,32,23,34,14,29,33,28);(5,8)(6,13)(7,18)(9,24)(10,25)(11,26)(12,28)(14,16)(15,33)(17,32)(19,21)(20,35)(22,36)(23,27)(29,30)(31,34);(5,8)(6,14)(7,19)(9,25)(10,24)(11,26)(12,29)(13,16)(15,33)(17,34)(18,21)(20,36)(22,35)(23,27)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=h,b^2=gh,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,e^b=ei,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	#72312195, #72312212, #72312219
maximal normal subgroup of	#72312195, #72312212, #72312219

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 273560);