#361718

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×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×#576	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3652 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312212, #72312252, #72312290
maximal normal subgroup of	#72312212, #72312252, #72312290

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268914);