#360228

order	512
nilpotency class	3
isoclinism	isoclinic class of #24295
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1580 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227326, #72312212, #72312238
maximal normal subgroup of	#72227326, #72312212, #72312238

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267424);