#368231

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1442 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307225, #72307310, #72307320
maximal normal subgroup of	#72307225, #72307310, #72307320

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 275427);