#7627247

order	512
nilpotency class	2
isoclinism	isoclinic class of #63011
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×D₄×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

8562 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)(5,6)(7,11)(8,9)(10,12)(13,14)(15,21)(16,22)(17,19)(18,23)(20,26)(24,27)(25,28);(6,9)(10,11)(13,15,18,17)(14,19,23,21)(16,24,26,25)(20,27,22,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,20)(15,24)(17,25)(18,26)(19,27)(21,28)(22,23);(5,7)(6,10)(8,12)(9,11)(13,17,18,15)(14,21,23,19)(16,25,26,24)(20,28,22,27);(13,16)(14,22)(15,24)(17,25)(18,26)(19,28)(20,23)(21,27);(13,14)(15,19)(16,20)(17,21)(18,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72304534, #72304584, #72304604
maximal normal subgroup of	#72304534, #72304584, #72304604

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7534443);