#6370299

order	512
nilpotency class	2
isoclinism	isoclinic class of #3532
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator	C₂²	C₂³×C₄²
socle	C₂³	C₂⁴×C₄
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2664 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=bf,c^a=ci,d^a=df,e^a=ei,c^b=cf,e^b=ei,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277495);