#376565

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2358
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=eh,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280175, #72280601, #72280718, …
maximal normal subgroup of	#72280175, #72280601, #72280718, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283761);