#519974

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#120
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fg,d^2=ghi,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^d=ef

Conjugacy classes & character table

140 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 427170);