#7625793

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=gh,e^2=h,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, 7532989);