#7627173

order	512
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂³	C₂⁶
commutator	C₂²	C₂⁵×C₄
prime radical	C₂×#3545	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

7182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

146 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534369);