#356234

order	512
nilpotency class	3
isoclinism	isoclinic class of #22455
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 263430);