#374119

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2476 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fhi,e^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,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, 281315);