#414977

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72281222, #72281264, #72281398, …
maximal normal subgroup of	#72281222, #72281264, #72281398, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 322173);