#7625781

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=g,e^2=g,b^a=bg,c^a=ch,f^c=fh,e^d=eh

Conjugacy classes & character table

170 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7532977);