#360498

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

990 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)(5,6,10,15,11,8,20,14)(7,16,18,12,19,27,31,21)(9,23,24,13,25,28,34,22)(17,30,32,29,33,26,35,36);(5,7,11,19)(6,12)(8,21)(9,17,25,33)(10,18,20,31)(13,26)(14,16)(15,27)(22,30)(23,29)(24,32,34,35)(28,36);(5,8)(6,11)(7,12)(9,22)(10,15)(13,25)(14,20)(16,18)(17,26)(19,21)(23,34)(24,28)(27,31)(29,32)(30,33)(35,36);(5,9,20,34,11,25,10,24)(6,13,15,28,8,22,14,23)(7,17,31,35,19,33,18,32)(12,26,27,36,21,30,16,29)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 267694);