#374117

order	512
nilpotency class	3
isoclinism	isoclinic class of #196110
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2479 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)(7,17)(8,13,22,30)(9,23)(10,25)(11,16)(12,19)(14,24)(15,26)(18,31,20,33)(21,35,36,29)(27,34,28,32);(1,2)(3,4)(5,7,11,9)(6,12,16,14)(8,18,22,20)(10,24,26,19)(13,27,30,28)(15,23,25,17)(21,34,36,32)(29,33,35,31);(2,4)(5,8)(6,13,16,30)(7,18)(9,20)(10,21)(11,22)(12,27,14,28)(15,29,25,35)(17,31,23,33)(19,32)(24,34)(26,36);(5,9,11,7)(6,14,16,12)(8,20,22,18)(10,19,26,24)(13,28,30,27)(15,17,25,23)(21,32,36,34)(29,31,35,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fh,d^2=i,e^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,h^c=hi,e^d=ei

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, 281313);