#357128

order	512
nilpotency class	3
isoclinism	isoclinic class of #3288
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1121 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)(5,6,10,17,12,18,11,16)(7,19,9,15,22,32,21,13)(8,23,25,14,26,30,27,31)(20,28,24,33,35,29,34,36);(1,2,4,3)(5,7,12,22)(6,13,18,15)(8,20,26,35)(9,11,21,10)(14,28,31,29)(16,32,17,19)(23,36,30,33)(24,27,34,25);(1,3,4,2)(5,8,10,25,12,26,11,27)(6,14,17,30,18,31,16,23)(7,20,9,24,22,35,21,34)(13,28,19,33,15,29,32,36);(5,9,11,7,12,21,10,22)(6,15,16,19,18,13,17,32)(8,24,27,20,26,34,25,35)(14,29,23,33,31,28,30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264324);