#374152

order	512
nilpotency class	3
isoclinism	isoclinic class of #24412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1706 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 281348);