#368514

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1363 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)(5,6,11,16,12,18,10,17)(7,19,21,13,22,14,8,23)(9,25,26,15,28,31,27,32)(20,29,34,30,35,36,24,33);(1,3,4,2)(5,7,12,22)(6,13,18,23)(8,11,21,10)(9,20,28,35)(14,17,19,16)(15,29,32,36)(24,26,34,27)(25,33,31,30);(5,8,10,7,12,21,11,22)(6,14,17,23,18,19,16,13)(9,24,27,20,28,34,26,35)(15,30,25,36,32,33,31,29);(5,9,12,28)(6,15,18,32)(7,20,22,35)(8,24,21,34)(10,27,11,26)(13,29,23,36)(14,30,19,33)(16,31,17,25)

Polycyclic presentation:

2^9|a^2=h,b^2=fh,c^2=g,d^2=i,e^2=i,f^2=i,g^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, 275710);