#374703

order	512
nilpotency class	3
isoclinism	isoclinic class of #25196
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
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

1219 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,g^b=gi,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, 281899);