#373863

order	512
nilpotency class	3
isoclinism	isoclinic class of #24283
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1270 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:

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

Polycyclic presentation:

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

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, 281059);