#385270

order	512
nilpotency class	3
isoclinism	isoclinic class of #30412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

969 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,c^b=cg,d^b=de,g^b=gi,g^c=gi

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, 292466);